450 
PROFESSOR STOKES ON THE COMMUNICATION OF VIBRATION 
resolved in a direction normal to the surface, V being a given function of t , 6, and <y ; 
then we must have 
^ =V, when r=c. 
(4) 
Another condition, arising from what takes place at a great distance from the sphere, 
will be considered presently. 
The sphere vibrating under the action of its elastic forces, its motion will be periodic, 
expressed so far as the time is concerned partly by the sine and partly by the cosine of 
an angle proportional to the time, suppose mat. Actually the vibrations will slowly 
die away, in consequence partly of the imperfect elasticity of the sphere, partly of com- 
munication of motion to the gas, but for our present purpose this need not be taken into 
account. Moreover there will in general be a series of periodic disturbances coexisting, 
corresponding to different periodic times, but these may be considered separately. We 
might therefore assume 
V = U sin mat U' cos mat , 
but it will materially shorten the investigation to employ an imaginary exponential in- 
stead of circular functions. If we take 
V= Ue imat , (5) 
where i is an abbreviation for —I, and determine <p by the conditions of the problem, 
the real and imaginary parts of cp and V must satisfy all those conditions separately ; 
and therefore we may take the real parts alone, or the coefficients of i or y / — 1 in the 
imaginary parts, or any linear combination of these even after having changed the arbi- 
trary constants which enter into the expression of the motion of the sphere, as the solu- 
tion of the problem, according to the way in which we conceive the given quantity V 
expressed. 
The function <p will be periodic in a similar manner to V, so that we may take 
<p=^e imat ( 6 ) 
As regards the periodicity merely, we might have had a term involving e~ imat as well 
as that written above ; but it will be readily seen that in order to satisfy the equation 
of condition (4) the sign of the index of the exponential in <p must be the same as in V. 
On substituting in (2) the expression for <p given by (6), the factor involving t will 
divide out, and we shall get for the determination of a partial differential equation 
free from t. Now \p may be expanded in a series of Laplace’s Functions so that 
'^ = '4 / o + 4 , 1 + '4 / 2+ (7) 
Substituting in the differential equation just mentioned, taking account of the funda- 
mental equation 
and equating to zero the sum of the Laplace’s Functions of the same order, we find 
(P-fy n . 2 dtyn 
dr 2 ' r dr 
n[n+ 1 ) 
r 2 
'pn+m 2 'p n = 0 . 
